Remarks on Scattering Properties of the Solution to a Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

Abstract

In this paper, we consider the Cauchy problem of Nonlinear Schrödinger equation \begin{align*} \left\{\begin{array}{ll}&i u_t+Δu=λ_1|u|^{p_1}u+λ_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=φ(x), \quad x\in\mathbb{R}^N, \end{array} \right. \end{align*} where $N\ge 3$, $0<p_1<p_2<\frac{4}{N-2}$, ${\lambda }_1$ and ${\lambda }_2$ are real constants. Using the methods in \cite{Cazenave2} and analyzing the interaction between the nonlinearity ${\lambda }_1|u{|}^{p_1}u$ and ${\lambda }_2|u{|}^{p_2}u$, we not only partly solve the open problems of Terence Tao, Monica Visan and Xiaoyi Zhang's \cite{Tao} but also obtain other scattering properties of the solutions.

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References in corpus (1)

• The nonlinear Schrödinger equation with combined power-type nonlinearities